This document describes fits from both state-space Ricker spawner-recruit models used fit in the Research Document (link once published online):

citation

Two state-space spawner-recruit models were fit in this paper that serve different purposes. First, a Ricker model with autoregressive (AR1) recruitment residuals was fit to estimate biological benchmarks (\(S_{gen}\), \(S_{MSY}\), and \(U_{MSY}\)), then a model with time-varying productivity (Ricker \(\alpha\) parameter) was fit in order to condition the biological submodel on recent population dynamics in the closed-loop forward simulation. These models were fit to each of the 9 CUs one by one, yielding 18 models to diagnose total. Data and code to reproduce the analysis and report is available in this GitHub repository, where you can see the R code to run the models, and the models themselves.

1 Diagnostics

We fit the spawner-recruitment model in a Bayesian estimation framework with Stan [@carpenter_stan_2017; @standevelopmentteamRstanInterfaceStan2023], which implements the No-U-Turn Hamiltonian Markov chain Monte Carlo algorithm [@hoffman2014] for Bayesian statistical inference to generate a joint posterior probability distribution of all unknowns in the model. We sampled from 4 chains with 2,000 iterations each and discarded the first half as warm-up. We assessed chain convergence visually via trace plots and by ensuring that \(\hat{R}\) (potential scale reduction factor; @vehtari2021rank) was less than 1.01 and that the effective sample size was greater than 400. Posterior predictive checks were used to make sure the model returned known values, by simulating new datasets and checking how similar they were to our observed data.

1.1 Trace plots

These should be clearly mixed, with no single distribution deviating from others (left column), and no chains exploring a strange space for a few iterations (right column).

1.1.1 NorthernYukonR.andtribs.

1.1.2 Whiteandtribs.

1.1.3 Pelly

1.1.4 Stewart

1.1.5 Nordenskiold

1.1.6 YukonR.Teslinheadwaters

1.1.7 MiddleYukonR.andtribs.

1.1.8 UpperYukonR.

1.1.9 Big.Salmon

1.2 ESS and \(\hat{R}\)

We hope minimum effective sample sizes are greater than 2000 and that \(\hat{R}\) are less than 1.05.

For the AR1 model:

CU ESS Rhat
NorthernYukonR.andtribs. 1022 1.002
Whiteandtribs. 525 1.010
Pelly 603 1.004
Stewart 1212 1.002
Nordenskiold 1674 1.003
YukonR.Teslinheadwaters 1110 1.003
MiddleYukonR.andtribs. 969 1.004
UpperYukonR. 1188 1.001
Big.Salmon 698 1.006

and the time varying model:

CU ESS Rhat
NorthernYukonR.andtribs. 70 1.050
Whiteandtribs. 572 1.008
Pelly 507 1.005
Stewart 48 1.091
Nordenskiold 1112 1.007
YukonR.Teslinheadwaters 813 1.003
MiddleYukonR.andtribs. 637 1.012
UpperYukonR. 62 1.056
Big.Salmon 465 1.016

We see some problems with the time varying model, and have explored various parametrizations and options to include semi-informative beta priors that did not improve diagnostics. We imagine these issues in the time-varying model are due to the extreme, pronounced decline in productivity, and the correlation between productivity and the stationary capacity prior (i.e., Ricker \(\beta\)).

**add a head(max(Rhat)) type thing to show what model*parms had problems?**